Question

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Starting at age 25, you deposit $2000 a year into an IRA account for retirement. Treat the yearly deposits into the account as a continuous income stream. If money in the account earns 5%, compounded continuously, how much will be in the account 40 years later, when you retire at age 65? How much of the final amount is interest? What is the value of the IRA when you turn 65? $(Round to the nearest dollar as needed.) How much of the future value is interest? $ (Round to the nearest dollar as needed.)

Answer 1

Value of IRA when you turn 65:

Future value (compounded continuously) =

yearly installment amount × [( e^rt - 1)/(e^r - 1)]

e = 2.718281828

Yearly installment amount : P = $2000

Rate of interest : r = 5% = 0.05

Number of payments made in 40 years : t = 40

= 2,000 × [(e^0.05×40 - 1)/(e^0.05 - 1)]

=2,000 × [(2.718281828² - 1)/(2.718281828^0.05 - 1)]

= 2,000 × [(7.3890560964 - 1)/(1.0512710964 - 1)]

=2,000 × 6.3890560964 / 0.512710964

= 249,226.4275589

= $ 249,226

The value of IRA when you turn 65 =$249,226

2nd part i.e. Interest amount

Total deposit amount =$ 2,000 × 40 years= $ 80,000

Total value at the end = 249,226

So interest = $(249,226 - 80,000)

=$ 169,226

The interest amount out of future value is  $ 169,226.

SUMMARY:

Value of IRA when you turn 65 = $249,226

Interest amount =$169,226